Math Table

Polynomial analogs of questions in number theory

Speakers: Hari Iyer

We discuss some analogs of hard, classical questions in number theory --counting prime numbers, solving equations-- which sometimes become easier when numbers are replaced by polynomials; this is known as a "function field analog," since the polynomials in question can also be viewed as functions on curves defined modulo a prime. Famous examples of such questions include the Riemann hypothesis and Fermat's Last Theorem. Time permitting, we may mention a curious application of this perspective to resolving a conjecture of Ramanujan on coefficients of modular forms.

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The statistics of increasing subsequences in biased random permutations

Speakers: Jonas Iskander

How does the cycle type of a random permutation affect the expected number of increasing subsequences of a given length? Recently, there has been growing interest in how the group structure on the symmetric group interacts with the natural ordering on the letters 1 through n. In a 2022 paper, Gaetz and Pierson studied the expected value of the number of length k increasing subsequences times a symmetric group character, and they found explicit formulas for these expected values in several special cases. In this talk, I will discuss how I generalized the results of Gaetz and Pierson to obtain a much broader class of explicit formulas for these expected values. In particular, my results imply explicit formulas for the expected number of length k increasing subsequences when a permutation is randomly generated according to a distribution that favors certain cycle types.

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Anosov Actions of PSL2(Z) on T^3

Speakers: Katherine Tung

Hurder (1992) showed that the natural action of SL2(Z) on the torus T^2 is not topologically rigid: there is a family of deformations that are not topologically conjugate to the standard action. By contrast, in dimensions higher than 2, the standard action of SLn(Z) on T^n is rigid, so there are no such families of deformations. There is a natural irreducible representation from PSL2(Z) to SL3(Z) inducing an action of PSL2(Z) on the 3-torus. Is this action rigid or not? We resolve this question by extending a technique of Hurder while preserving the elementary nature of his argument. This talk is based off of a Northwestern REU project I worked on with Tanner Leonard, Nathan Louie, and Paul Shin.

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Graduate Student Panel

Speakers: NA

Merrick Cai - Mathematics graduate student at Harvard 

Ricky Li Economics - graduate student at MIT 

Hanna Mularczyk - Mathematics graduate student at MIT

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Statistical Inference of Finite-Population Galton-Watson Branching Processes

Speaker: Ivan Specht

Abstract: The Galton-Watson process is a stochastic branching process in which the offspring distribution of each vertex is an independent and identically distributed (i.i.d.) N-valued random variable following a prespecified distribution ξ. Here, we consider a modification of the process in which the set of vertices in any realization is sampled from a fixed, finite set. In this modification, which we refer to as a finite-population Galton-Watson process, offspring distributions are no longer i.i.d., and must be re-normalized based on the remaining population size after each draw. We show that for large population size, the probability of a finite-population Galton-Watson process containing a fixed subtree admits a tractable approximation with tight error bounds for several choices of ξ. Finally, we discuss the importance of approximating this probability for modeling infectious disease transmission, a common application of the Galton-Watson process.

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The Pursuit of Quasi-excellence: Using complete local rings to study the prime spectra of quasi-excellent local integral domains

Speaker: AnaMaria Perez

Abstract: Complete local rings are well understood due to a powerful theorem from Cohen.The completion of a local ring with maximal ideal $M$ can be defined using the $M$-adic metric. Characterizing the relationship between a local ring and its completion allows us to deduce results about the original ring. In particular, we characterize the relationship between quasi-excellent local domains and their completions. We prove that $T$, a complete local ring of characteristic 0 with maximal ideal $M$, is the completion of a local quasi-excellent integral domain if and only if no nonzero integer of $T$ is a zero-divisor and $T$ is reduced. We also provide necessary and sufficient conditions for countable local quasi-excellent domains. One notable application of our results is that there is no bound on how non-catenary a quasi-excellent integral domain can be, a result that was previously unknown. Finally, we use these results to motivate the study of formal fiber rings and provide a characterization for completions of local integral domains where we can control formal fibers of countably many principally generated ideals. We may extend these characterizations to completions of countable local domains, quasi-excellent local domains, and excellent local domains.

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Putnam Post Mortem

Speaker: Noam Elkies

Abstract: Join our own Professor Noam Elkies to discuss a selection of this year’s Putnam problems.

Bender-Knuth Billiards in Coxeter Groups

Speaker: Eliot Hodges

Abstract: Abstract: In this talk, which will be accessible to undergraduates of all levels, we discuss the way in which Schützenberger's famous promotion map--a permutation of the linear extensions of a poset--can be generalized to a combinatorial sorting operator on Coxeter groups. We also give a description of this Coxeter-theoretic promotion in terms of (noninvertible) Bender-Knuth toggles, which generalize Bender-Knuth involutions to Coxeter groups. In addition to classifying many of the cases where promotion will "sort" (such Coxeter groups are dubbed futuristic), we describe an alternate, more geometric way of thinking about the problem, in which we have a black hole and a beam of light traveling through (or reflecting off of) a series of one-way mirrors. (This talk includes joint work with Barkley, Defant, Kravitz, and Lee.)

3D Tic-Tac-Toe

Speaker: Yanni Raymond

Abstract: If you ever spent an afternoon of your childhood playing tic-tac-toe, you may have noticed that the game typically ends in a draw. Indeed, with optimal play from both players, the game will be a draw. But what about tic-tac-toe played on different board sizes, or in more than two dimensions? This talk will introduce pairing strategies, strategy stealing, and the Erdös-Selfridge theorem to resolve planar tic-tac-toe and introduce hypercube tic-tac-toe.

Peg Solitaire on Graphs with Large Diameter

Speaker: Daniel Sheremeta

Abstract: Peg solitaire is a classical table game played on a variety of different boards. In this talk, we explore the generalisation of peg solitaire to arbitrary graphs, which was proposed as early as 1994 by Moulton. Based on research that I did at Harvard this summer, the talk’s focus is on the solvability (i.e. whether all of the pegs can be removed) of graphs with a large number of edges. I hope to keep things extremely accessible, assuming only a familiarity with the basic vocabulary of graph theory.

Combinatorics of Crystal Graphs

Speaker: Jack Mann

Abstract: Crystal graphs are a useful tool to have when working with representations of quantum groups. While quantum groups are complex algebraic structures, we can represent elements in their representations as Young tableaux. With this method, we can develop a more abstract understanding of its algebraic structure using counting strategies. In this talk, we examine the crystal graph corresponding to the symplectic Lie algebra and look for patterns in the arrangement of crystals and their weights, aiming to find the dimensions of subspaces in the corresponding representation.

Game of Cones: Jeu de Taquin and Simplicial Complexes

Speaker: Dora Woodruff

Abstract: In this talk, we will explore a connection between topology and combinatorics. To every partially ordered set, objects often studied by combinatorialists, one can assign a simplicial complex and study its topological properties. For example, a theorem of Björner says that whenever the partially ordered set is a lattice, the resulting complex is homeomorphic to a ball. We'll start by discussing this classical theory, and then explore a new direction that brings in another combinatorial player: Young tableaux. Based on some research I did this summer, a meta-theme of this talk will also be that it's not always terrible news when your conjectures are wrong!

Job Panel

Please join us for a panel discussion about what careers and jobs you can pursue with a background in mathematics. Our panelists are 

• Sarah Minucci, Senior Scientist - Mathematical Modeler at Applied BioMath .

• Aiden Carey, Software Engineer at Klaviyo.

• Heemyung Hwang, Software Engineer at Zagaran.

• Vasily Ilin, Data Engineer at Google.

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Do you know a proof of the fundamental theorem of algebra?

Speaker: Cliff Taubes, Harvard University

Abstract: The fundamental theorem of algebra (d’Alembert’s theorem) says that any monic polynomial of degree n has n roots counting multiplicity.  There are lots of proofs—but which is the simplest?  And, what do you need to know to prove it?

Finding Balance in Chaos: Approximating Orthogonal Polynomials on Julia Sets

Speakers: Madison Shirazi

Abstract: Equilibrium measures describe optimal charge distributions on sets while Julia sets partially characterize the iterative behavior of polynomials. We will construct the equilibrium measures of certain Julia sets in the complex plane and then define the orthogonal polynomials relative to these equilibrium measures. Surprisingly, certain orthogonal polynomials turn out to be the iterates of the generating polynomials of the Julia sets. We will see how we can approximate these polynomials using Gram-Schmidt and study different refinements of the Gram-Schmidt algorithm.

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Finding Simple Models of Complex Objects: From Regularity Lemmas to Algorithmic Fairness

Speakers: Sílvia Casacuberta Puig

Abstract: In recent years, algorithms are increasingly informing decisions that can deeply affect our lives. A major concern that arises is whether prediction algorithms are fair across different subpopulations. The notions of multiaccuracy and multicalibration were proposed by Hébert-Johnson et al. in 2018 as mathematical measures of algorithmic fairness.

Our starting point is the observation that multiaccuracy is exactly what is given by the Regularity Lemma, which is an older result in computational complexity shown by Trevisan, Tulsiani, and Vadhan in 2009 that has many important implications in different areas. By formalizing this observation, we then ask: If we start with a multicalibrated predictor instead, what versions of these implications do we obtain? Through the lenses of algorithmic fairness, we are able to cast the notion of multicalibration back into the realm of complexity theory and obtain stronger and more general versions of these theorems.

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Lean into Theorem Proving

Speakers: Janna Withrow, Mark Pekala, William Hu, and Peter Chon

Abstract: The programming language Lean has ushered in a new era in formal mathematics, providing a robust platform for validating proofs with exceptional reliability. Grounded in a rigorous, type-theoretic framework, Lean empowers mathematicians to engage with avant-garde mathematical inquiries and experiment with innovative proof methodologies. In our presentation, we will illuminate the mechanics of Lean and its myriad applications, emphasizing its potential to elevate our comprehension of intricate mathematical notions. We will showcase its versatility in formalizing diverse mathematical structures by highlighting renowned theorems that have been meticulously verified using Lean. Furthermore, we will exemplify its capacity to unveil fresh insights, as illustrated by Peter Scholze's recent contributions. Lastly, we will recount our firsthand experiences with employing Lean in Math 161, highlighting its merit as a pedagogical instrument for cultivating a deeper understanding of proof theory. We hope you will join us as we delve into the unique opportunities offered by Lean to examine, authenticate, and intensify our grasp of mathematics.

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Non-compact Calabi-Yau manifolds from Calabi-Yau cones

Speaker: Benjy Firester

Abstract: Calabi-Yau manifolds are complex Kähler manifolds satisfying algebraic criteria that implies the existence of a unique Ricci-flat metric (in the compact case), which is the so-called the Calabi Conjecture as proven by Yau, giving rise to the eponymous name. Yau’s proof supplies a large class of interesting geometries, including most known examples of Einstein manifolds, new tools in string theory, and many geometric and topological corollaries, however the abstract existence technique means we have very little understanding of explicit formulas for the metric. In the non-compact case, existence and uniqueness is rarely known, but we can better understand the geometry of Calabi-Yau metrics. In this talk, I will discuss recent constructions of complete non-compact Calabi-Yau manifolds with motivations to understand compact manifolds, especially in degenerate limits. 

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Quaternions and abcs

Speaker: Raphael Tsiamis

Abstract: Can you solve a 3x3 equation that encodes quaternions into differential geometry?

Quaternions have three imaginary units i,j,k; they are, to complex numbers, what complex numbers are to the reals. These are the building blocks for smooth and complex manifolds. The next step are hyperkähler manifolds, resembling the quaternions; they are full of interesting properties, but difficult to construct. Kronheimer identified such structures coming from differential equations. In a simple case, studying the solutions of a 3x3 equation reveals the geometry of a hyperkähler manifold through interesting pictures.

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Graduate School Panel Discussion

Please join us for a panel discussion about graduate school! We'll cover questions about the application process, how to choose a graduate program, planning courses that help with grad school, and more!

Panelists:

Daniel Abdulah, MIT Planetary Science

Anne Larsen, MIT Mathematics

Lucy Liu, Harvard Applied Mathematics

Phillip Nicol, Harvard Biostatistics

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Explicit Formulas for Permutation Pattern Character Polynomials

Speaker: Jonas Iskander

Abstract: Given a permutation σ on k letters and a permutation π on n letters, one can ask how many times the ordering pattern σ occurs in substrings of π. For fixed σ, Janson, Nakamura, and Zeilberger showed that the number of pattern occurrences in a uniformly random permutation π follows a normal distribution. However, it is natural to ask how the distribution of numbers of pattern occurrences interacts with the group structure on permutations. Recently, Christian Gaetz, Laura Pierson, and Christopher Ryba introduced the family of permutation pattern character polynomials, which measure the expected value of the number of pattern occurrences times a symmetric group character. In this talk, I will describe how I extended their work to derive explicit formulas for certain permutation pattern character polynomials.

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The Inhomogeneous TASEP and Evil-Avoiding Permutations

Speaker: Katherine Tung

Abstract: The asymmetric simple exclusion process (ASEP) is a family of models for interacting particles. A particular type of ASEP -- the inhomogeneous TASEP -- is only partly understood, and is related to Schubert polynomials and a family of permutations called "evil-avoiding." In this talk, I will discuss the inhomogeneous TASEP, pattern-avoiding permutations, and some related research I did last summer at the Duluth REU.

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Putnam Competition Postmortem

Speaker: Noam Elkies, Harvard University

Abstract: Solutions, outlines, and/or mathematical context will be given for some of the problems from the Putnam exam on December 3, starting with problems that most lend themselves to discussion.

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Pushing a Camel through the Eye of a Needle

Speaker: Maira Khan

Abstract: I will present some ideas from the proof of Gromov's Non-Squeezing Theorem with emphasis on a more dynamical view. I hope to give some intuition about symplectic "width" and deformations of manifolds in phase space. 

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The Ax-Grothendieck Theorem

Speaker: Hari Iyer

Abstract: Results in mathematics with simple statements often have surprising proofs. This talk will present such an example, applying structures from set theory to solve a cute problem about polynomial maps between vector spaces.

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Breakdown of the Einstein field equations and space-time singularities

Speaker: Oswaldo Vazquez

Abstract: General Relativity (GR) is our best theoretical framework for describing large scale phenomena such as supernovae, rotating black holes, planetary orbits, etc. The theory models space and time as a 4-dimensional semi-Riemannian manifold M and posits that the force of gravity is manifested through the curvature of M. The equations of GR are powerful but not perfect as they can lead to singular solutions containing blow-ups of geometric invariants. The most famous GR singularity is that which occurs at the center of a (static) spherically symmetric black hole. In theory, there can also exist singularities that are not necessarily hidden under an event horizon (these pathological singularities are called "naked") . It is then natural to ask: what are the conditions that allow space-time to have such singularities? In this week's Math Table, I will give a heuristic overview on how one may tackle the singularity formation problem as well as recent progress I have made with Dr. Puskar Mondal.

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Extended Promotion and Quasi-tangled Labelings 

Speaker: Eliot Hodges

Abstract: In 2022, Defant and Kravitz introduced extended promotion (denoted ∂ ), a map that acts on the set of labelings of a partially ordered set. Extended promotion is a generalization of Schützenberger's promotion operator, a well-studied map that permutes the set of linear extensions of a poset. A remarkable fact about promotion is the following: if L is a labeling of an n-element poset, then ∂^{n-1}(L) is a linear extension. This result allows us to regard promotion as a sorting operator on the set of all labelings of a finite poset. In this talk, I will introduce extended promotion and discuss some research on promotion I did at the Duluth REU. In particular, the talk will conclude with an outline of the enumeration of the quasi-tangled labelings of a poset as well as the statement of some open problems and further directions of inquiry.

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